The Flutter Problem:
Modeling Unstable Aeroelastic Vibrations in Rocket Fins
Vincent San Miguel
MATH 450: Mathematical Modeling
The Pennsylvania State University
1 Background
The purpose of this project was to study the phenomenon known as aeroelastic flutter and its behavior and effects on rocket fins. This was accomplished by reviewing existing data and previouslyconducted studies, developing equations based in advanced mechanics of materials theory and aeroelastic theory, creating Finite Element Analysis (FEA) and Computational Fluid Dynamics (CFD) models, and analyzing solutions in MATLAB. Each tool was utilized with the goals of analyzing aerodynamic and structural behavior during fin flutter and creating a differential equation that could effectively model the mode shapes of a fin while vibrating at resonant frequency.
1.1 Aeroelastic Flutter
Flutter is defined as the “dynamic instability associated with the interaction of aerodynamic, elastic, and inertial forces” [1]. The three types of flutter are stall flutter, pitchbending flutter, and bendingtorsion flutter. The most likely type of flutter for a properly constructed rocket – and thus the focus of this investigation – is bendingtorsion flutter, in which two natural modes of a fin’s oscillations interact through another shared resonance. In fin bendingtorsion flutter, the diversion of air around the fin creates forces that cause both bending and torsion. Continuous aerodynamic loads cause a coupling between these moments in which the energy from bending is deflected into torsion and back again, leading to a positive feedback loop that, if left underdamped, could result in shearing and structural failure. Flutter also generates lift, and can thus cause the rocket to become dynamically unstable. Even in airplane wings, stiffness criteria based on flutter requirements is in many instances the critical design criteria.
As rocket fins don’t often have large mechanical dampers, rockets must rely on careful construction to avoid the effects of flutter. Below a certain velocity, air flow around the fin acts as a rather efficient way of damping any vibrational energy that might occur during flight. Above this velocity however, air flow will exponentially amplify perturbations and lead to underdamped oscillations. During flutter, the fin will vibrate at or near the natural mode of frequency characteristic of the material and geometry. These vibrations cause displacements throughout the fin, whose corresponding stresses can lead to shear and ultimately complete structural failure of the fin.
2 A Theoretical Framework
In order to quantitatively analyze flutter, a mathematical approach was taken to study the variables and relationships involved in the phenomenon. Several methods were used to develop initial equations that could model simplified systems and provide useful information for the more complicated fin geometries.
2.1 Mathematics of Vibrations
To begin studying the mathematics behind flutter, the model was simplified to that of a laterally vibrating beam. By utilizing advanced mechanics of materials theory, a fourth order partial differential equation of motion for a forced lateral vibration of a nonuniform beam can be obtained:


(2.1)

By making the assumptions of a uniform beam undergoing free vibration, the equation becomes


(2.2)

2.1.1 Solving the Clamped Uniform Beam Free Vibration Equation
A solution for equation (2.2) can be acquired analytically through the method of separation of variables. The values of lateral displacement and velocity are specified as and at , so the initial conditions become


(2.2a)

Substituting into equation (2.2) and rearranging leads to


(2.2b)

Equation (2.2b) can be rewritten as two equations:


(2.2c)

Finally, the solutions for equation (2.2c) are expressed as


(2.3)

where was determined by using the auxiliary equation . The function is known as the characteristic function of the beam and determines lateral displacement along the xaxis direction of the beam. The natural frequency of vibration can be determined from equation (2.2c) as


(2.4)

where is the xdirection distance between the two endpoints of a beam undergoing bending due to vibration.
Four boundary conditions are needed in order to find a unique solution for equation 2.2b. For a fixedend beam with a freely vibrating end, these are:
Free end:


(2.5a)

Clamped end:


(2.5b)

By rewriting equation (2.3) and for the boundary conditions, the function for the mode shape of a transverse vibration of a beam can be determined as


(2.6)

2.2 Mathematics of Aeroelasticity
Through the use of aeroelastic theory and wind tunnel and flight test data, equations estimating flutter velocity and shear velocity have been produced. The following equations are taken from NACA Report 685 [2].


(2.7)

Where represents the speed of sound, represents the fin material’s shear modulus, is the pressure ratio, is the aspect ratio, is the fin thickness, and is the root chord length. Shear velocity is experimentally approximated to be around .
3 Computational Methods
After a mathematical framework was established, a visual analysis of flutter vibrations was generated using FEA and CFD software. This allowed for the quantitative analysis of the causes of fin flutter and the effects it has on the structure.
3.1 Finite Element Analysis
Mode shapes – the displacement at every point of the geometry for a corresponding natural frequency – were determined through the use of FEA. In addition, Von Neumann stresses were also calculated.
3.1.1 Methodology
Analyses were conducted using CATIA v5 software. A 3D model of a fin was constructed based on dimensions and materials taken from a flighttested rocket in which flutter velocities were calculated [3]. The fin was modeled after the clipped delta fin constructed for “Rocket #2” by Ray Dunakin and had a root chord of 8.25 in, a span of 5.25 in, and a tip chord of 5.25 in. It was made of 6061T6 Aluminum, and was modeled as such in CATIA. After this, a frequency analysis was conducted using CATIA’s FEA software package. Two scenarios were analyzed: one in which the fin was epoxied to the airframe and one in which it was held by a fin bracket.
Figure 1: Epoxied Fin – 6061 Al
Figure 2: Bracketed Fin – 6061 Al
3.1.2 Results and Analysis
Natural modes of frequency for the structures were determined, and corresponding stresses and mode shapes were calculated. Unfortunately, FEA frequency analysis cannot account for external forces and pressures being applied to the model. As such, stresses, strains, and natural frequencies are overestimated when compared to those of a model experiencing external forces. Frequency analysis also fails to account for fracture, and so the calculated strains tend to be unreasonably high. Nevertheless, these analyses proved useful in displaying the mode shapes of fin geometry under resonant frequency, as well as the locations of highest stress and strain. Analyzing the two different scenarios also quantitatively justified the use of fin brackets in missiles and rockets by showing how they can greatly reduce the chances of flutter and stresses on the fin, as mode one resonant frequency occurs at a level over 25 times that of a regularly epoxied fin with minimal fin displacement.
Figure 3: Mode 1 Frequency Analysis with Mesh
Figure 4: Epoxied Fin  Mode 1 Vibration (84.73 Hz)
Figure 5: Bracketed Fin  Mode 1 Vibration (2113 Hz)
Figure 6: Epoxied Fin  Mode 3 Vibration (455.68 Hz)
Figure 7: Bracketed Fin  Mode 3 Vibration (51810 Hz)
Figure 9: Bracketed Fin  Mode 8 Vibration (294837 Hz)
Figure 8: Epoxied Fin  Mode 8 Vibration (1431 Hz)
3.2 Computational Fluid Dynamics
A computational approach was taken to determine the characteristics of airflow around the fin responsible for flutter. Velocity fields and pressure gradients were calculated to study the effects of wind speed and pressure on a fin.
3.2.1 Methodology
Using Ansys Workbench, a two dimensional CFD model of a fin with the dimensions stated previously and a finite flow field was constructed. A mesh containing 87,497 nodes was then generated. Using Ansys FLUENT, the fluid was modeled as air, with a density of 1.225 kg/m^{3} and a viscosity of 1.7894 x 10^{05} kg/ms. The structure was once again modeled as 6061 Aluminum, with a density of 2,719 kg/m^{3}. The inlet air flow was modeled as laminar at a velocity of 160 m/s and standard temperature and pressure. Solution methods involved a SIMPLE scheme, and momentum was solved using a secondorder upwind scheme. In order to ensure accuracy, FLUENT ran for 4,241 iterations until the solution converged within an acceptable error of 1 x 10^{6}.
Figure 0: 2D Flow Field Mesh
3.2.2 Results and Analysis
The velocity field resulting from the fin is expected of a laminar flow and is a useful visualization tool for establishing changes in flow velocity around the fin. These flow fields can indicate where forces responsible for flutter are acting on the fin.
Figure 11: Velocity Field (inlet velocity = 160 m/s in xdirection)
Figure 12: Velocity Field Contours
Figure 13: Velocity Field (magnified)
Static pressure contours show the magnitude and distribution of external forces caused by air flow on the fin. A pressure buildup of 1.44 x 10^{4} Pa can be seen at the vertex where the leading edge of the fin and the wall meet, indicating the location of highest external forces on the structure. A small area of negative pressure can also be seen coming off the leading tip.
Figure : Static Pressure Contours
Total pressure contours in Figure 15 reveal the formation of a negative pressure differential behind the fin responsible for creating additional drag.
Figure 15: Total Pressure Contours
Although not capable of modeling flow around a vibrating structure, CFD analysis was useful for determining the distributions and magnitudes of external forces throughout the edges of the fin. This information can be used in helping create a mathematical model of flutter.
4 Developing a Mathematical Model
With a mathematical framework in place, FEA and CFD analyses completed, and data from thirdparty wind tunnel and flight tests available, a mathematical model for determining mode shapes could begin to be constructed.
4.1 MATLAB Modeling of Flutter Equations
To begin modeling fin vibrations, the system was simplified to a 2^{nd} order ordinary differential equation representing a driven harmonic oscillator:


(4.1)

where is the damping coefficient. If , the system is unstable and will experience exponentially growing oscillations – the case for vibrations above flutter velocity. Table 1 displays the possible values for damping coefficients and their corresponding effect on the behavior of vibrations.

Damped behavior ()


Undamped


Unstable

Table : Damping Coefficients
Figure 16: y'' = y'  y^{3}, with and boundary conditions [0 1]
The equation was then separated into two first order differential equations that could be solved using MATLAB’s numerical ODE solver, ode45.
Figure 16 shows the resulting exponentially increasing oscillations of a negative damping coefficient. Additional ODEs were modeled in MATLAB, but none contributed any additional information beyond what was shown by equation (4.1).
A fourth order partial differential equation was then attempted to be solved numerically using MATLAB’s pdesolver function. Unfortunately, due to unfamiliarity with the function, a solution was unable to be properly established despite numerous attempts. However, based on the mathematical results derived analytically in Section 2, it can be hypothesized with some level of support that the PDEs modeling the mode shapes for fin flutter are at least of the fourth order.
5 Conclusions
Although computational analysis provided useful insight into fin flutter, an effective mathematical description of mode shapes could not be reached. Due to the inherently complicated nature of aeroelastic vibrational analysis, producing a higher order partial differential equation that could accurately model fin flutter proved to be too difficult for the scope of this project. Nevertheless, certain goals pertaining to fin flutter analysis were accomplished.
Mathematical analysis of cantilevered freely vibrating beams showed that the mathematics of even very simple models could get complicated very quickly. They were however useful in establishing a theoretical framework for building a finbased vibrational model from a fourth order partial differential equation.
Finite Element Analysis allowed for the visualization of mode shapes for fin geometries and the distribution of stress throughout the structure when vibrating at resonant frequency. It revealed the complex nature of the problem by showing the complicated displacements of a vibrating fin. It also provided an adequate justification for the use of fin brackets, which were shown to have a first mode resonant frequency over 25 times that of a regularly epoxied fin.
CFD analysis allowed for the determination of pressure distributions on the fin resulting from fluid flow, as well as changes in flow properties as a result of fin geometry. Pressure buildup on the structure was identified, and the resulting distribution of forces along the fin edges was qualitatively analyzed.
MATLAB solutions were calculated based on ordinary differential equations modeling forced harmonic oscillators. These equations acted as a simple case for modeling the effects of undamped vibrations on a fin. The solutions, however, were not sophisticated enough to effectively model three dimensional mode shapes, as problems with utilizing MATLAB’s PDE solver prevented an obstacle to finding probable solutions. Despite not arriving at a mathematical equation that could model fin flutter, effective analysis of its causes and effects has been shown to be possible through the use of FEA and CFD techniques.
6 Resources
[1] Hodges, Dewey H., and G. Alvin. Pierce. Introduction to Structural Dynamics and Aeroelasticity.
Cambridge [England: Cambridge UP, 2002. Print.
[2] Theordosen, Theordore, A Theoretical and Experimental Investigation of the Flutter Problem,
NACA Report 685, 1940.
[3] McDonald, Duncan. Fin Flutter Estimator. Excel Datasheet.
